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In-Range Farthest Point Queries and Related Problem in High Dimensions

Authors Ziyun Huang, Jinhui Xu

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LIPIcs.ICALP.2022.75.pdf
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Author Details

Ziyun Huang
  • Department of Computer Science and Software Engineering, Penn State Erie, The Behrend College, USA
Jinhui Xu
  • Department of Computer Science and Engineering, State University of New York at Buffalo, NY, USA

Funding

  • Xu, Jinhui: The research of this author was supported in part by NSF through grant IIS-1910492 and by KAUST through grant CRG10 4663.2.

Cite AsGet BibTex

Ziyun Huang and Jinhui Xu. In-Range Farthest Point Queries and Related Problem in High Dimensions. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 75:1-75:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.75

Abstract

Range-aggregate query is an important type of queries with numerous applications. It aims to obtain some structural information (defined by an aggregate function F(⋅)) of the points (from a point set P) inside a given query range B. In this paper, we study the range-aggregate query problem in high dimensional space for two aggregate functions: (1) F(P ∩ B) is the farthest point in P ∩ B to a query point q in ℝ^d and (2) F(P ∩ B) is the minimum enclosing ball (MEB) of P ∩ B. For problem (1), called In-Range Farthest Point (IFP) Query, we develop a bi-criteria approximation scheme: For any ε > 0 that specifies the approximation ratio of the farthest distance and any γ > 0 that measures the "fuzziness" of the query range, we show that it is possible to pre-process P into a data structure of size Õ_{ε,γ}(dn^{1+ρ}) in Õ_{ε,γ}(dn^{1+ρ}) time such that given any ℝ^d query ball B and query point q, it outputs in Õ_{ε,γ}(dn^ρ) time a point p that is a (1-ε)-approximation of the farthest point to q among all points lying in a (1+γ)-expansion B(1+γ) of B, where 0 < ρ < 1 is a constant depending on ε and γ and the hidden constants in big-O notations depend only on ε, γ and Polylog(nd). For problem (2), we show that the IFP result can be applied to develop query scheme with similar time and space complexities to achieve a (1+ε)-approximation for MEB. To the best of our knowledge, these are the first theoretical results on such high dimensional range-aggregate query problems. Our results are based on several new techniques, such as multi-scale construction and ball difference range query, which are interesting in their own rights and could be potentially used to solve other range-aggregate problems in high dimensional space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Farthest Point Query
  • Range Aggregate Query
  • Minimum Enclosing Ball
  • Approximation
  • High Dimensional Space

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